where \(\frac{\mathop {d\sigma _\delta }}{\mathop {d\mathcal {B_\delta }}}\)  is the fully differential Born cross section for the partonic channel \(\delta \)  and \(\mathcal {H}\)  implements the kinematic cuts applied to the Born
phase space \(\mathcal {B}\) . \(\mathcal {F}\)  denotes the multiple
emission function which, for additive observables such as the
angularities considered in this paper, is simply given by
\(\mathcal {F}(L) = e^{-\gamma _E R^\prime }/\Gamma (1+R^\prime )\) ,
with \(R^\prime (L)=\partial R/\partial L\)  and
\(R(L)=\sum _{l\in \delta } R_l(L)\) . The ratio of
parton-distribution-functions (PDFs) \(\mathcal {P}\)  takes into account
the true initial-state collinear scale. The soft function
\(\mathcal {S}\)  implements the non-trivial aspects of colour evolution.
The collinear radiators \(R_l\)  for the hard legs \(l\)  were computed
in [1]} for a general observable \(V\)  scaling for the
emission of a soft-gluon of relative transverse momentum \(k_t^{(l)}\) 
and relative rapidity \(\eta ^{(l)}\)  with respect to leg \(l\)  as
\(V(k)=\mathopen {}\mathclose {\left(\frac{k_{t}^{\mathopen {}\mathclose {\left(l\right)}}}{\mu _Q}\right)}^{a}e^{-b_{l}\eta ^{\mathopen {}\mathclose {\left(l\right)}}}d_{l}\mathopen {}\mathclose {\left(\mu _Q\right)}g_{l}\mathopen {}\mathclose {\left(\phi \right)}\,.\) 
